general relativity - Inconsistency with partial derivatives as basis vectors?

I have been trying to convince myself that it is consistent to replace basis vectors $\hat{e}_\mu$ with partial derivatives $\partial_\mu$. After some thought, I came to the conclusion that the basis vectors $\hat{e}_\mu$ were ultimately just symbols which represent what we think of as arrows, so it is not a problem to use a different symbol. The only requirement is that one can manipulate the $\partial_\mu$ in the same way as the $\hat{e}_\mu$.

However, raising/lowering indices seems to create an inconsistency. In switching our representation of the basis vectors, we make the substitutions:

$$\hat{e}_\mu \rightarrow \partial_\mu$$

$$\hat{e}^\mu \rightarrow dx^\mu$$

However, while we previously could write $\hat{e}^\mu=g^{\mu \nu}\hat{e}_\nu$, we fail to be able to write the same relationship in the new representation:

$$dx^\mu \neq \partial^\mu =g^{\mu \nu} \partial_{\nu}$$

My questions are:

• Have I done something invalid here?


• If not, is it just an unwritten rule that one should never try to raise an index of a basis vector?


• What is the motivation to write basis vectors as partial derivatives or differentials (for the tangent or cotangent space) as opposed to just writing some other symbol? Do we actually need the properties of a derivative or differential in our basis vectors? I am aware that the $\partial_\mu$ resemble the expression $\frac{\partial\vec{r}}{dx^\mu}$ which is a natural choice for the basis vectors $\hat{e}_\mu$, but the differentials seem to come out of nowhere.

Answer

Raising and lowering indices in a vector is not a valid operation. Basis vectors are no exception. While $x_\mu=g_{\mu\nu}x^\nu$ is a valid operation, $\hat e^\mu=g^{\mu\nu}\hat e_\nu$ is not. The reason is that in the first case you are dealing with the components of a vector, and in the second case you are dealing with a vector itself.

Let me elaborate. Given a vector $\hat X$ $$\hat X=x^\mu\hat e_\mu$$ you can lower the index $\mu$ in $x^\mu$ through $$x_\mu\equiv g_{\mu\nu}x^\nu$$

That is: raising and lowering indices is an operation that is defined for the components of a vector (or covector).

The index $\mu$ in $\hat e_\mu$ is not a vector index; it just labels the different basis vectors. You cannot raise/lower this index, because $\hat e_\mu$ does not denote the components of any vector. The operation $$\phantom{\color{red}{\text{NO!}}}\qquad\hat e^\mu\equiv g^{\mu\nu}\hat e_\nu\qquad\color{red}{\text{NO!}}$$ is a meaningless operation.

The same thing can be said about covectors. Given a covector $\tilde X$ $$\tilde X=x_\mu\tilde e^\mu$$ you can raise the index in $x_\mu$. But you cannot lower the index in $\tilde e^\mu$, because that index does not denote the components of a covector; it just labels the different basis covectors.

Most importantly, while $\hat e_\mu$ is a basis of the space of vectors, and $\tilde e^\mu$ is a basis for the space of covectors, these objects are not related through $$\phantom{\color{red}{\text{NO!}}}\qquad\hat e^\mu= g^{\mu\nu}\tilde e_\nu\qquad\color{red}{\text{NO!}}$$ or any similar relation.

In short: you can raise/lower indices when those indices denote the components of an object - either a vector or a covector - but you cannot raise/lower the indices of the bases of vectors/covectors, because those indices do not denote the components of anything. They are just labels.

However, see Musical isomorphism.

I hope that at this point, you are still with me. Given an arbitrary vector $\hat v$ (like $\hat X$ or $\hat e_\mu$), and a certain function $f$, we can define the action of $\hat v$ on $f$ as follows: we define $$\hat e_\mu[f]\equiv \frac{\partial f}{\partial x^\mu}\in\mathbb R$$ and we extend this through linearity: if $\hat v=v^\mu \hat e_\mu$, then $$\hat v[f]=v^\mu\frac{\partial f}{\partial x^\mu}\in\mathbb R$$

I'm not going to discuss why this new operation is useful. But let me stress that this operation is something new, something that you might have never seen before: now vectors can act on functions! In any case, useful or not, this new operation motivates us to consider the following convenient notation: we will write $\hat \partial_\mu$ instead of $\hat e_\mu$: $$\hat \partial_\mu\equiv \hat e_\mu$$

With this, our equation from before now becomes $$\hat\partial_\mu[f]=\frac{\partial f}{\partial x^\mu}$$

Note that we are using the same symbol, $\partial$, with two different meanings: on the one hand, it denotes a basis vector, and on the other hand, it denotes a partial derivative. The usual thing we do is to drop the distinction: we just write $\partial_\mu$ for both, and let context decide what the symbol means.

In the same vein, we usually use the symbol $\mathrm dx^\mu\equiv\tilde e^\mu$. That is, we denote the basis of covectors by the symbol $\mathrm dx^\mu$. It's just notation.

Let us now move on to the gradient. We define the covector $\mathrm d f$ as the covector that has $\frac{\partial f}{\partial x^\mu}$ as components: $$\mathrm d f=\frac{\partial f}{\partial x^\mu}\tilde e^\mu$$ or, using our new notation, $$\mathrm d f=\partial_\mu f\,\mathrm dx^\mu$$

You can raise and lower the $\mu$ index in $\frac{\partial f}{\partial x^\mu}$, because this index denotes the components of a covector. In this sense, you could say that you can raise/lower the $\mu$ index in $\partial_\mu$, whenever this symbol denotes a derivative. But you cannot raise/lower the $\mu$ index in $\hat \partial_\mu$, whenever this symbol denotes a basis vector (for the same reason you cannot raise/lower the $\mu$ index in $\hat e_\mu$).

In short: the objects $\partial_\mu$ and $\mathrm dx^\mu$ replace the old notation $\hat e_\mu$ and $\tilde e^\mu$, but they denote the exact same object: they are a basis for the space of vectors and covectors. This means that you cannot raise/lower their indices. On the other hand, the object $\partial_\mu f$ denotes the components of the covector $\mathrm df$, and as such, you can raise/lower its index.